An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit

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• 作者：Antonio Giorgilli ; Simone Paleari ; Tiziano Penati
• 刊名：Annales Henri Poincare
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：16
• 期：4
• 页码：897-959
• 全文大小：760 KB
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• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Dynamical Systems and Ergodic Theory
  Quantum Physics
  Mathematical Methods in Physics
  Relativity and Cosmology
  Elementary Particles and Quantum Field Theory
  
• 出版者：Birkh盲user Basel
• ISSN：1424-0661

文摘

We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β 1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.